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[12 points] Jerry" Truck Company determined that the distance traveled per truck per year is normally distributed, with mean of 50 thousand miles and standard ...

Question

[12 points] Jerry" Truck Company determined that the distance traveled per truck per year is normally distributed, with mean of 50 thousand miles and standard deviation f [2 thousand miles What proportion (aka fraction or percentage) of trucks can be expected to travel between 34 thousand and 50 thousand miles per year? What percentage of trucks can be expected to travel either less than 30 thousand or" more than 60 thousand miles in year? How many miles will be traveled by at least 80

[12 points] Jerry" Truck Company determined that the distance traveled per truck per year is normally distributed, with mean of 50 thousand miles and standard deviation f [2 thousand miles What proportion (aka fraction or percentage) of trucks can be expected to travel between 34 thousand and 50 thousand miles per year? What percentage of trucks can be expected to travel either less than 30 thousand or" more than 60 thousand miles in year? How many miles will be traveled by at least 80 percent of the trucks? What would be your answers (0 (a), (b) and (€) if the standard deviation were I0 thousand miles?



Answers

The mean lifetime of a certain tire is 30,000 miles and the standard deviation is 2500 miles. a. If we assume the mileages are normally distributed, approximately what percentage of all such tires will last between 22,500 and 37,500 miles? b. If we assume nothing about the shape of the distribution, approximately what percentage of all such tires will last between 22,500 and 37,500 miles?

In this question. We have information about the average number of miles driven by a licensed driver. So this the average that were given is 14,000 90 miles, driven by the average driver with a standard deviation of 3500. So part of the question is asking us if somebody drives 16,000 miles, what is their Z score? So let's review the formula for finding a Z score. Z is equal to X minus X bar over s where X is the individual data value. X bar is the mean and s is the standard deviation. So we can simply substitute each of these pieces of information into the formula and calculators. He's so Z will equal 16,000 minus 14,090 divided by 3500. And when we compute that subtraction then divided by 3500 we find that the Z score is 0.55 In other words, a person who drives on average 16,000 miles is 0.55 standard deviations above average for the amount of drivers, uh, the amount of miles driven by licensed drivers. Part V says, What if the driver only drives 10,000 miles. So what is the Z score for somebody who only drives 10,000 miles? So somebody who's driving less than average. So we would expect when we put this into our formula on we do our calculations, we would expect this answer to come out to be negative because the amount of miles driven is below average. And in fact, when we do that when we do our calculations, it turns out that this is a Z score that is negative 1.17 So this person who drives on average 10,000 miles drives 1.17 standard deviations less than the average driver imports see, and has us sort of switching gears, working in the other direction. So it wants to know. Then what would the Z scores be? Not Does he scores? What would the data values be for people who have driven Brownie, whatever amount of miles where we end up with either 1.6 0.5 or zero as R Z score. So we just have to work backwards. So let me bring that formula back down. We write it over here, so we have it so X minus X bar over us. So we're going to substitute the pieces of information that we know into this formula and saw for what we don't know. So these are Z scores, so we'll start with the 1st 11.6 is equal to X. That's the piece. I don't know. That's what I'm looking for, minus the mean divided by the standard deviation. And then we need to use algebra to work backwards to solve this equation. So we will multiply both sides of this equation by 3500 and then when we're done with out, we're gonna add so 3500 times 1.6 is 5600. And that leaves us with our data value, minus the mean And then to finish solving this equation, we're gonna add the mean to both sides, and we find out that the data value that we are looking for is 1 19,000 690. Okay, so for 0.5, we're gonna follow exactly the same procedure except 0.5 will be substituted in for see, we have the same mean and the same standard deviation. So the right side of the formula doesn't change it all. And so when we follow that procedure, same thing multiply both sides by 3500 than at 14,090. We find that this X this data value is equal to 15,000 840. And that makes sense because we have a number that is above the mean but only a little bit of off the meat on Li like half a standard deviation on Li like 1717 50 above the meat. Okay, I'm and finally our last one. What if Z is equal to zero and we don't even need to do any calculation here of C equal to zero means what? That tells us that we're looking for the mean what data value is zero standard deviations above or below the mean And that is, in fact the mean. So that would mean that the data value would have to be 14,000. So somebody who drives 14,090 miles on average drives the mean amount of miles. In other words, there's the score would be syrup and no Cappy, no calculations are involved. Just a little bit of reasoning

So in this question, were given some information about the average miles people drive with the standard deviation and asked to find some Z scores and evaluate some Z scores. So here's are given information were given that X bar is 14,090. The average amount of miles driven is 14,090 with a standard deviation of 3500. And it would be good for us to review what the formula is for a Z score, since we have to to find a Z score. So the formulas x minus X bar over s where X is individual data value explores the sample mean and s is the sample standard deviation. All right, so question A says what is the Z score? If somebody drives on average 16,000 miles, So 16,000 his ex here's army. And in our standard deviation, we simply have to plug these values into our formula and divide. So when we do that Calculation 16,000 minus 14 90 divided by 3500 is 0.5 around two decimal places. 0.55 Eso somebody who drives 16,000 is driving over the national average. So there's he scores positive, but it's not. Ah, whole standard deviation over the average is just a little. So just 0.55 standard deviations above average. Okay, question be says, what is a Z score for somebody who drives Oops, not explore X for somebody who drives 10,000 miles. So somebody who drives less than average Um, let me just clean up that notation that we're calculating a Z score. All right, so C equals that in 10,000 minus 14,090 divided by 3500. Now, this is going to give us a negative z score because we are dealing with a knave rich Doran X value that is below 14,090. From when we put that into our calculator, subtracting first and then dividing, we get negative 1.1 seven. So this person drives a little bit more than one standard deviation, but below average. All right, questions see then is saying Okay, what are the X values or what are the data values that are associated with three different Z scores? So first is a Z score of 1.6. So now I know the Z, but I don't know X were so we're kind of working in a different direction. So it's still substitute everything I know into the formula and then simplify. So 1.6 equals the data value acts that I don't know minus 14,090 divided by 3500. Now, to solve this equation, to find X, we need to work backwards, and we need to multiply both sides of this equation by 3500. So when I do that, I get 5600 is equal to X minus 14,090 Then to find acts, I'm simply gonna add 14,090 over to the other side of this equation. And I find out that the speed associated with the Z score of 1.6 is 19,690 Next. Part of the question says, OK, if the Z score is negative 0.5, what would the X value be? So we're gonna follow a similar procedure, except we're gonna let the Z score equal negative 0.5 and again, we don't know what the data value is, but we do know what the mean and what the standard deviation are following a similar procedure. We're going to multiply 3500 on both sides of the equation, and we get negative. 1750 is equal to X minus 14,090. And then when we add 14,090 to the other side, we find out that the miles are the average amount of miles driven by this driver is 12340 12,000, 340 months. All right, The last part of the question says Okay, what if the Z score is zero? And let's think about what that means. If positive means it's above average, a negative means it's below average. Zero would mean it's exactly average. So in other words, the person would have to drive exactly the amount of the average, so we don't even have to do any calculations. The Z score would be 15,090 because they this person drives neither above nor below the average no standard deviations above or below the

Welcome to memory. In the current problem. We are given the lifetimes the lifetimes of certain types of one years tires. Okay, so the sudden by himself touches lifetime most fires and we are also given that X follows a normal distribution with new equals one like kilometer, been lax kilometer and is equal to 10,000 no reliability. The question asking is what percentage? Well what is a bit of fires? What percentage of barriers between 85 1000 kilometers and then kilometers? Okay, then lack kilometers. So that is equivalent to asking this. Now, if we do a standardization, we get Divided by 10,000. Now, if we simplify this, we get probability now 100 -85 gives us 15, correct, This is it value and this is zero. So we will have -1.5, less than zero, less than zero. So in standard normal distribution like this, This will be zero And this will be -1.5, correct. Somewhere we're here. So, but what happens is we have 1.5 given over here that too this area completely. So the table gives okay, the table gives then less than area, correct. So this area. So what can we do? We want this area because this area and this area will be cool, correct. So we'll be right probability of That. Less than zero Minour probability of that is the -1 point faith, Correent. So probability of said less than zero Minour probability of zed greater than 1.5, which is equals two, probability zero, less than zero minus one minus probability zero less than 1.5. Because our table gives value less than that. So for 1.5 if I bring the diagram, little town, no 1.5 We have this value, correct? So we and for zero we have this value. It's we know already minus one plus zero point I don't want 9332, correct? So this and this will give 0.933 to minus 0.5 because 0.5 minus one is minus zero point right? So we get 0.4332 which Will be 43.32%. So 43.2% of all tires will have uh huh. Lifetime between 85,000 km two then black kilometers.


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